What’s the relationship between gain (dB) and power (watts)?

+1 millercarbon, you're spot on.

Yes, and thanks.

Awesome question and one I’ve been contemplating myself.

and awesome answer MC, thanks to the op and the og

None. :) 

Amps tend to, by convention but not requirement, have around 28 dB of gain, regardless of their power rating. :)  This is about 20x input voltage.

That is, put in 0.1V peak to peak input signal and you should get 2V peak to peak on the output.

Of course, gain only works so long as you don't exceed the output limits, whose absolute limit in a linear amp is by the power supply rails.  So, if your rails are +- 20V, that's your peak output.  With 20x gain (around 28 dB) this means your maximum input voltage is 1v peak to peak before clipping.

Thanks everyone.  I like the gas pedal vs engine analogy.  But does a higher initial gain ultimately mean anything for the output of the amplifier (like higher possible SPL in dB for a given speaker)?